You Get What You Share: Incentives for a Sharing Economy

The Thirty-Third AAAI Conference on Artiﬁcial Intelligence (AAAI-19)

Sreenivas Gollapudi

Kostas Kollias

Debmalya Panigrahi

Google Research

Duke University

Abstract

their individual resources. The platform allows the agents to list and search for available resources, which enables them to identify partners and form sharing groups. More specifically, in workforce and educational applications, there are groups that work to complete a task or a project, while in ride sharing and room sharing applications, the notion of a group appears when agents get together to use a provided resource, e.g., a ride or a house.

In recent years, a range of online applications have facilitated resource sharing among users, resulting in a significant increase in resource utilization. In all such applications, sharing one’s resources or skills with other agents increases social welfare. In general, each agent will look for other agents whose available resources complement hers, thereby forming natural sharing groups. In this paper, we study settings where a large population self-organizes into sharing groups. In many cases, centralized optimization approaches for creating an optimal partition of the user population are infeasible because either the central authority does not have the necessary information to compute an optimal partition, or it does not have the power to enforce a partition. Instead, the central authority puts in place an incentive structure in the form of a utility sharing method, before letting the participants form the sharing groups by themselves. We first analyze a simple equal-sharing method, which is the one most typically encountered in practice and show that it can lead to highly inefficient equilibria. We then propose a Shapley-sharing method and show that it significantly improves overall social welfare.

A natural goal is to partition the users into sharing groups that maximize the overall utility or social welfare of the system. One may model this as an optimization problem, where a centralized authority computes the optimal partition of users. However, in typical environments, the information necessary for computing this partition is hard to obtain due to privacy issues or due to the sheer size of the population. Moreover, the central authority has only limited control over the participants, and often cannot enforce the partition. Hence, in this paper, we study the following question:

Consider a population of individuals, who can form sharing groups and share their individual resources with group members. What is the right way to distribute the welfare produced by each sharing group among its members so that the population self-organizes in a socially optimal manner? The

semantics of distributing welfare are different for each application. For instance, in online learning, utility is given in the form of course credits that depend on how students contributed to group projects. In the ride-sharing application, using a vehicle has some value that is the same for all agents and monetary transfers reward those providing resources and extract payments from those utilizing resources to adjust utilities accordingly.
We model instances of this problem as games that we call resource sharing games. The most natural way to split the produced welfare among the group’s members is to divide it equally. In fact, this is the method that naturally comes into play when there are no monetary or credit transfers in our applications of interest, i.e., everyone simply gets access to the resources available in the group. However, we show that this method can perform quite sub-optimally with respect to overall social welfare. We then propose an incentive structure derived from the well-known Shapley value in the economics literature (Shapley 1953), and show that the quality of the resulting outcomes is signiﬁcantly improved.

Introduction

The sharing economy (Belk 2014) is a term used to describe the modern trend of using online platforms and applications to increase the value of certain resources (such as goods, services, or skills) by enabling the sharing and reuse of these resources. Carpooling and ride sharing applications (e.g., Uber, Lyft, RideWith) allow people to better utilize commuting resources (cars, fuel, etc.), lodging and room sharing applications (e.g., AirBnB) allow users to share their space when not using it, online learning platforms (e.g., Coursera, Udacity, edX) allow students to share knowledge and collectively learn, and workforce applications (e.g., Upwork, Amazon Mechanical Turk) allow workers to collaboratively complete requested projects by contributing complementary skills. In all such applications, there are natural limits on the extent to which the resources can be shared. In carpooling, each car can ﬁt a small number of people; in lodging, each room or house has a ﬁxed capacity; in crowsourced projects and online courses, there is a prescribed size for working groups. In effect, this means that the agents need to be organized in groups of limited size so as to effectively share

c

Copyright ꢀ 2019, Association for the Advancement of Artiﬁcial

In our model, each participant owns a subset of all possi-
2004 ble resource types and, when a group is formed, everyone in the group has access to all resource types that at least one of them owns. An important observation is that the resulting social welfare function is not submodular, but supermodular. (This is in contrast to an existing body of work that studies utility games for submodular welfare functions, e.g., (Vetta 2002; Gairing 2009; Marden and Wierman 2013).) Informally, this means that a group produces more utility than the sum of its individuals. While this is often the case in practice, it can be theoretically problematic because the only stable solution is typically one where all the agents join the same group. However, in real-world applications (we give some examples below), it is impractical to have very large groups. So, we will adopt a model where groups are subject to a cardinality constraint, and also show that it is possible to enforce such a constraint through incentives alone.
(in the form of a utility-sharing method) that will lead the agents to efficient outcomes.
We begin by studying the most natural and widely used utility sharing method, which dictates that each agent has utility equal to the number of resources available in her sharing group. This is equivalent to equally splitting the social welfare produced by a group among its members and, hence, we call it equal-sharing (EQUAL). For instance, in online course projects, EQUAL would mean all students in a group will get the same course credit, i.e., the problems that the group has solved. In the carpooling application, EQUAL means all agents get to take the ride and split any costs (e.g., fuel) equally. We show that, for any reasonable equilibrium concept, there is an equilibrium partition of agents into sharing groups that is k times worse than the optimal. We also explain that this inefficiency with respect to the optimal is the worst possible for any partition, independent of whether it is an equilibrium or not. In light of this observation, we seek an alternative utility-sharing method, which results in better equilibria. The method we propose is derived from the Shapley value (Shapley 1953) and, hence, we call it Shapley-sharing (SHAPLEY). SHAPLEY aims to improve on the main shortcoming of EQUAL, which is that it rewards agents based on their contribution to improving the social welfare by sharing their resources. Below we present our results for the various equilibrium concepts, which show that SHAPLEY leads to better equilibria than EQUAL. Nash equilibrium. The Nash equilibrium (NE) is the most established equilibrium concept in the literature. An NE in our setting is a partition such that no agent can improve her utility by unilaterally deviating to a different group. It is clear that the set of equilibria of a resource sharing game depends on the utility-sharing method being used. The price of anarchy (POA) (Koutsoupias and Papadimitriou 2009) is used as a metric that quantifies the performance of a utilitysharing method by calculating the worst-case ratio of the social welfare in the optimal partition to the social welfare in an NE. As we mentioned earlier, the POA of EQUAL is k. Unfortunately, the POA remains Θ(k) for SHAPLEY as well. In fact, we show that this will be the case for any natural utility-sharing method. However, we observe that the examples that yield this universal POA lower bound are unnatural as they assume agents will form groups of very small sizes, even when they are allowed to form larger sharing groups with more resources available. To get past these unnatural pathological cases, we consider the following two equilibrium concepts.

Applications

First, we describe two concrete real world examples of our resource sharing games:

• Massive Open Online Courses – Suppose we are inter-

ested in partitioning the set of students into groups to complete an assignment comprising multiple problems. From a social perspective, we would prefer a partition of students such that every group in the partition completes as many problems as possible. Each student on the other hand, wishes to maximize the course credit she will receive. The goal of the central authority (teacher) is to provide the right incentives (in the form of a grading scheme) to the students so that effective groups grow organically and in a decentralized fashion.

• Carpooling – In a carpooling system, agents form groups and repeatedly share rides to and from work during rush hour. In this setting the resources of the game correspond to car rides from location to location and welfare is generated for each agent who takes a ride. From a social perspective, we wish that the commuting resources are put to good use. On the other hand, agents wish to optimize their individual rewards. Providing the appropriate incentives and inducing the organic formation of efﬁcient sharing groups is important in achieving the social objective.

Several other applications such as crowdsourcing and labor markets similarly ﬁt our proposed model.

Our Contributions

We consider a setting where a population of agents seeks to share resources in groups of size at most k. The value of k, the desired group size, is application specific (e.g., in (Davis 2009), values of 4, 5, and 6 are prescribed as appropriate for educational applications, while similar such constraints are natural in carpooling and other applications). Each group generates social welfare, which is then distributed as utility to the agents who generated it. We study this setting as a game, where the agents self-organize into groups. Each agent wishes to optimize her own utility share. We assume the role of the central authority, who cannot enforce the final sharing groups, but who can establish an incentive structure
Strong Nash equilibrium. The strong Nash equilibrium (SNE) is an NE that is robust to deviations by coalitions of agents (Aumann 1959). This implies that a partition is an SNE when there is no subset of agents who can coordinate and deviate to other groups (or start groups on their own), thereby improving their individual utilities. The strong price of anarchy (SPOA) is the direct analog of the POA for the SNE. We prove that SHAPLEY results in a significant improvement of the SPOA, which is brought down to 2 (from k for EQUAL). Unfortunately, this equilibrium concept has the well-known drawback that it is not guaranteed to exist in many settings. Indeed, we give examples of resource shar-

2005 ing games in our setting that have no SNE under SHAPLEY.
Toward the goal of characterizing equilibria with more desirable outcomes in the context of unilateral deviations, we focus on an equilibrium concept that we deﬁne here, and which we argue is the one that best predicts the behavior of selﬁsh agents in our settings. We call this new concept a

balanced Nash equilibrium.

Each agent i in group G, on the other hand, receives an individual utility u_i(G). Utility-sharing is budget-balanced, i.e.,

u_i(G) = U(G).

∑

i∈G

We now deﬁne the equilibrium concepts that we will study.

Balanced Nash equilibrium. We deﬁne a balanced Nash

equilibrium (BNE) as an NE such that all groups, except possibly one, have k agents, thereby ensuring balance in terms of their size. In many applications (such as in carpooling and online courses), this is an end in itself. In other settings, minimizing the number of groups subject to cardinality constraints is a desirable outcome. Note that merging small groups can only increase social welfare as more resources are added to the group and more agents beneﬁt from them. We will also show that the BNE has the following properties:

Deﬁnition 1. A partition G is a Nash equilibrium (NE) when for every agent i in group G, it is the case that u_i(G) ≥ u_i(G⁰∪ {i}) for every other group G⁰∈ G . In other words, no agent can improve her utility by switching to any other sharing group unilaterally.

As mentioned earlier, we also use the coalition-resistant notion of strong Nash equilibrium.

Deﬁnition 2. A partition G is a strong Nash equilibrium (SNE) if no coalition of agents S ⊆ N can deviate (move to another group, including forming a new group) to form a new partition G ⁰such that u_i(G⁰) > u_i(G) for every agent i ∈ S, where G⁰and G are the respective groups of agent i in G ⁰and G . Intuitively, an SNE is a partition such that no coalition of agents can coordinate and move between groups in a way that beneﬁts every member of the coalition.

• Ease of enforcement – It is easy for the central authority to slightly modify the SHAPLEY method so that every NE of the resulting game is a BNE of the original game, i.e., a BNE can be easily enforced.

• Existence – A BNE is always guaranteed to exist.
Finally we introduce the notion of a balanced Nash equi-

librium (BNE).

• Efﬁciency – The loss of efﬁciency in a BNE (we call this

the balanced price of anarchy (BPOA) by analogy) is at

most 3 in the SHAPLEY scheme (as against the BPOA of k of EQUAL).

Deﬁnition 3. A partition is a balanced Nash equilibrium (BNE) if it is an NE such that all groups, except possibly one, have k agents.

Finally, we perform simulations which show that the
SHAPLEY method improves on EQUAL in realistic settings, with respect to three metrics that measure our desiderata for forming sharing groups: performance (measured by the social welfare of the partition), participation (measured by the number of sharing groups of size k), and fairness (measured by the quality of the weakest group).
We will show existence of this equilibrium and demonstrate that it can be easily enforced by the central authority by modifying the utility-sharing method. We also note that BNE outcomes are desirable since they enforce groups of large size, where sharing is maximized and social welfare is increased.
Having introduced our three equilibrium concepts, we are ready to deﬁne the metrics that we use to measure the performance of a utility-sharing method.

Preliminaries

In this section, we describe our model and introduce the notation. There is a set of agents N and a set of resource types P. Let P ⊆ P denote the set of resource types that agent i ∈ N poⁱssesses. Agents self-organize into groups so as to share their available resources. We denote the set of groups by G . The social welfare of a group G ∈ G is:
Deﬁnition 4. The price of anarchy (POA), the strong price of anarchy (SPOA), and the balanced price of anarchy (BPOA)

are the worst case ratios of the optimal social welfare to the social welfare of an NE, SNE, and BNE respectively, i.e.,

max_GU(G )

min_{G is an NE}U(G )

max_GU(G )

min_{G is an SNE}U(G )

max_GU(G )

min_{G is a BNE}U(G )

S

(

POA = SPOA = BPOA =

...

|
P|·|G|,

i

if |G| ≤ k

i∈G

U(G) =

(1)

0,

otherwise where k is the cardinality parameter that represents the maximum group size. The expression encodes the fact that every agent in a group beneﬁts from the presence of each resource available in the group. Note that this deﬁnition assumes that all resources have unit value. This assumption is without loss of generality since resources with non-uniform utility can be modeled by multiple resources with equal utility. The overall objective is to form sharing groups that maximize social welfare

Equal Sharing

The most natural way of sharing utility from resources in a group is to have every agent in the group receive utility equal to the value of the resources. Note that this is equivalent to equally splitting the welfare generated by the group among its participants and, hence, we call this utility-sharing

method equal-sharing (EQUAL).

U(G ) =

U(G).

∑

G∈G

2006

Theorem 5. The POA, SPOA, and BPOA are all k under

EQUAL.

• If agent i owns a resource of type p, then she gets 1 unit of utility for p.
• If agent i does not own a resource of type p and the group

Proof. Consider the following setting. There is only one resource type. There are k agents who own the resource (we call them strong agents) and k · (k − 1) agents who do not (we call tham weak agents). The optimal solution G ^∗places every strong agent in a different group and adds k−1 weak agents to each group. Then all agents have access to the unique resource type in the game and the social welfare is U(G ^∗) = k². Now consider the partition G where all strong agents are in the same group and all other agents are arbitrarily partitioned into groups of size k. We will show that this partition is an SNE, an NE, and a BNE. We ﬁrst argue that it is an SNE: Note that all strong agents receive the maximum possible utility of 1, and hence they have no incentive to deviate from this partition in any way. Furthermore, the weak agents have no way to improve their utility above 0, since if any subset of them were to join the strong group, its cardinality would increase beyond k leading to total utility of 0. This proves that G is an SNE. It follows that it is an NE as well, since every SNE is also an NE. Observing that the number of groups is equal to the ratio of the number of agents and the cardinality parameter, i.e., k, proves that G is also a BNE. Note that U(G ) = k. Then we get that the POA, the SPOA, and the BPOA are all at least k.
It is not hard to see that this is actually the worst possible ratio for any possible partition (not only for equilibria). For each resource an agent owns, the least social welfare earned by that resource is 1, in cases when only that agent receives utility for it. The maximum possible is k, in cases when she shares it with another k −1 agents in her group.

We note that this strong lower bound on the SPOA suggests that such inefﬁcient equilibria can appear in many settings of interest, with weaker notions of stability. This includes situations where two agents can swap their group memberships or where a small number of agents can coordinate and deviate to a new group.

has n agents who have that resource and share it with her,

n

then she gets

units of utility for that resource. The re-

n+1
1

maining

n+1

units are shared uniformly among those who shared resource p with her (so each of them gets extra utility).

1

n(n+1)

• If the size of the group is larger than k, each agent is penalized for a unit of utility per resource type.

SHAPLEY is connected to the the standard definition of the Shapley value. The standard definition prescribes that each agent is rewarded with the expected increase she causes to the group welfare function U(·) over a uniformly random order of arrival of the agents in the group. In our setting the exact Shapley value we get when plugging in U(·) as the welfare functions is hard to work with and get a closed form expression. Our SHAPLEY instead is equivalent to the Shapley value under the following modification: Agents get utility equal to the expected increase they cause to the welfare function after removing the capacity constraint (i.e. the Shapley value with infinite capacity), but they also share a penalty equal to their cardinality if the size of the group is larger than k. This penalty is split evenly across agents since they all contribute equally to bringing the group size to larger than k.
SHAPLEY is an instantiation of the Shapley-value for the modified setting described above and, hence, guarantees the existence of an NE in every resource sharing game (Kollias and Roughgarden 2011). Intuitively, SHAPLEY is engineered to address the drawbacks that caused EQUAL to be very inefficient. It incentivizes strong agents to move to groups where they can offer more help and also incentivizes them to spread over multiple groups (since the bonus of an agent who shares is inversely proportional to the number of owners of the same resource type in the group). At the same time, it breaks symmetry in utility earned by the agents, allowing them to break away from very inefficient partitions that have groups of size k.

Shapley Sharing

In light of the poor performance of EQUAL in terms of the three inefficiency metrics, we seek a utility-sharing method that addresses the main shortcoming of EQUAL, which is that agents who significantly increase social welfare by sharing with their peers are not rewarded for it. The Shapley value (Shapley 1953) from the economics literature provides a method for splitting social welfare among those generating it. We propose applying the Shapley value in our setting, which yields a natural way of partially transferring utility from the agents who do not have a resource to the agent sharing it with them. In this section, we analyze the performance of Shapley-sharing (SHAPLEY) and show that it results in a significant improvement in the quality of equilibria.

Nash Equilibria

By equivalence to Shapley value, we get that an NE is guaranteed to exist in our game for SHAPLEY utility sharing (Kollias and Roughgarden 2011). Indeed, we show in our full version (Gollapudi, Kollias, and Panigrahi 2018) that better-response dynamics always converges to an NE.
However, it turns out that any utility-sharing method, including SHAPLEY, that has an agent split her utility with those who shared resources with her has a POA of Θ(k). We call such methods share-rewarding. Due to space limitations, we omit the proof, which appears in our full version (Gollapudi, Kollias, and Panigrahi 2018).